Lehmann-Scheffé Theorem

Definition

The Lehmann-Scheffé Theorem is a fundamental result in statistics that provides conditions under which an estimator is considered admissible and also establishes that the best unbiased estimator is unique if it exists. This theorem connects the concepts of completeness and admissibility, emphasizing how a complete sufficient statistic can help identify the optimality of estimators.

5 Must Know Facts For Your Next Test

1. The theorem states that if an estimator is unbiased and based on a complete sufficient statistic, it is admissible.
2. The Lehmann-Scheffé theorem provides a way to find the best unbiased estimator among a class of estimators when certain conditions are satisfied.
3. The existence of a unique best unbiased estimator hinges on the completeness of the underlying sufficient statistic.
4. If a statistic is not complete, then there may be multiple unbiased estimators with the same minimum variance.
5. The theorem highlights the interplay between completeness and admissibility, making it a cornerstone for understanding optimal estimation in statistics.

Review Questions

• How does the concept of completeness relate to the criteria for admissibility as outlined by the Lehmann-Scheffé Theorem?
  • Completeness plays a crucial role in determining admissibility according to the Lehmann-Scheffé Theorem. An estimator that is based on a complete sufficient statistic and is unbiased is deemed admissible. This means that if you have a complete statistic, you can confidently assert that there isn't another estimator that would outperform your current one in terms of risk across all parameter values.
• Discuss how the Lehmann-Scheffé Theorem informs the search for the best unbiased estimator in statistical practice.
  • The Lehmann-Scheffé Theorem guides statisticians in identifying the best unbiased estimator by establishing that such an estimator exists if based on a complete sufficient statistic. It directs attention to finding these key statistics, as they ensure uniqueness and optimality. By understanding this theorem, practitioners can streamline their efforts in creating estimators with minimal variance while maintaining unbiasedness.
• Evaluate the implications of the Lehmann-Scheffé Theorem on developing statistical methodologies in real-world applications.
  • The implications of the Lehmann-Scheffé Theorem extend deeply into statistical methodologies used in real-world applications. By emphasizing the importance of completeness and admissibility, it encourages researchers to construct estimators that are not only unbiased but also efficient. This has practical consequences in various fields, including economics and medicine, where making accurate predictions based on data can significantly impact decision-making processes.

Related terms

Complete Statistic: A statistic is complete if no non-trivial function of the statistic has an expected value of zero for all parameter values.

Admissibility: An estimator is admissible if there is no other estimator that performs better in terms of risk across all parameter values.

Sufficient Statistic: A statistic is sufficient if it captures all necessary information from the data about the parameter being estimated.